C = q / m
is the same for all the objects. The Coulomb force and Newton's gravitational force are completely analogous between any two objects, except that the forces have opposite directions.
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The electromagnetic force between any two objects we calculate from the the electric and the magnetic fields produced by them, using the Hendrik Lorentz force (1895) formula
F = q E + q v × B
and the Oliver Heaviside (1888) formula for the magnetic field of a point charge:
B = μ₀ / (4π) q v × r₀ / r²,
where r₀ = r / |r| is the unit vector to the direction of the vector r.
Oliver Heaviside in 1893 suggested that Newton's gravity should be extended by a magnetic gravity field which is perfectly analogous to electromagnetism.
Let us assume that the magnetic field B is nothing more than the Lorentz transform of the Coulomb force. Then there must be an analogous gravitomagnetic field B_g for Newton's gravity force, and we must be able to calculate the effects of gravitomagnetism using equations which are completely analogous to electromagnetism.
This is indeed the case. According to Wikipedia, general relativity implies the gravitomagnetic laws for weak fields and slowly moving objects.
In this blog we are claiming that general relativity can be derived from special relativity and Newton's gravity. Now we got more evidence for our claim. Frame dragging close to a rotating black hole is explained by gravitomagnetism, which is a consequence of special relativity and Newton's gravity force.
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